Select The Correct Answer.What Is The Sum Of The First Eight Terms In This Series?${ 2 + 10 + 50 + 250 + \cdots }$A. 156,250 B. 195,312 C. 317 D. 97,656
When dealing with a series of numbers, it's essential to identify the pattern and understand how each term is generated. In this case, we have a series that starts with 2 and each subsequent term is obtained by multiplying the previous term by 5.
The Series: 2 + 10 + 50 + 250 + ...
To find the sum of the first eight terms in this series, we need to understand the pattern and calculate each term individually.
Calculating Each Term
The first term is 2. To find the second term, we multiply 2 by 5, which gives us 10. The third term is obtained by multiplying 10 by 5, resulting in 50. We continue this process to find the subsequent terms.
| Term | Value |
|---|---|
| 1 | 2 |
| 2 | 10 |
| 3 | 50 |
| 4 | 250 |
| 5 | 1250 |
| 6 | 6250 |
| 7 | 31250 |
| 8 | 156250 |
Calculating the Sum
Now that we have the first eight terms, we can calculate the sum by adding them together.
Sum = 2 + 10 + 50 + 250 + 1250 + 6250 + 31250 + 156250 Sum = 195312
Conclusion
Based on the calculations, the sum of the first eight terms in the series is 195,312.
Answer
In the previous section, we discussed a series of numbers that starts with 2 and each subsequent term is obtained by multiplying the previous term by 5. We calculated the sum of the first eight terms in the series and found that it is 195,312. In this section, we will answer some frequently asked questions related to the series and its sum.
Q: What is the pattern of the series?
A: The pattern of the series is obtained by multiplying the previous term by 5. This means that each term is 5 times the previous term.
Q: How do I calculate each term in the series?
A: To calculate each term in the series, you can use the formula: term = previous term * 5. For example, to find the second term, you would multiply the first term (2) by 5, resulting in 10.
Q: What is the formula for the sum of the series?
A: Unfortunately, there is no simple formula for the sum of this series. However, we can use the formula for the sum of a geometric series to find the sum. The formula for the sum of a geometric series is: S = a * (r^n - 1) / (r - 1), where a is the first term, r is the common ratio, and n is the number of terms.
Q: How do I use the formula for the sum of a geometric series?
A: To use the formula for the sum of a geometric series, you need to know the first term (a), the common ratio (r), and the number of terms (n). In this case, a = 2, r = 5, and n = 8. Plugging these values into the formula, we get:
S = 2 * (5^8 - 1) / (5 - 1) S = 2 * (390625 - 1) / 4 S = 2 * 390624 / 4 S = 195312
Q: What is the sum of the first 10 terms in the series?
A: To find the sum of the first 10 terms in the series, we can use the formula for the sum of a geometric series. Plugging in the values a = 2, r = 5, and n = 10, we get:
S = 2 * (5^10 - 1) / (5 - 1) S = 2 * (9765625 - 1) / 4 S = 2 * 9765624 / 4 S = 9765624
Q: What is the sum of the first 15 terms in the series?
A: To find the sum of the first 15 terms in the series, we can use the formula for the sum of a geometric series. Plugging in the values a = 2, r = 5, and n = 15, we get:
S = 2 * (5^15 - 1) / (5 - 1) S = 2 * (30517578125 - 1) / 4 S = 2 * 30517578124 / 4 S = 15258789062
Conclusion
In this article, we discussed a series of numbers that starts with 2 and each subsequent term is obtained by multiplying the previous term by 5. We calculated the sum of the first eight terms in the series and found that it is 195,312. We also answered some frequently asked questions related to the series and its sum.
Answer
The correct answer is B. 195,312.